A351980 - OEIS

%I #9 May 20 2022 09:27:00

%S 1,6,84,126,140,210,490,525,686,875,1404,1456,2106,2184,2288,2340,

%T 3432,3510,5460,6760,7644,8190,8580,8775,9100,9464,11466,12012,12740,

%U 12870,13650,14300,14625,15808,18018,18468,19110,19152,20020,20672,21450,22308,23712

%N Heinz numbers of integer partitions with as many even parts as odd conjugate parts and as many odd parts as even conjugate parts.

%C The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

%F Closed under A122111 (conjugation).

%F Intersection of A349157 and A350943.

%F A257992(a(n)) = A344616(a(n)).

%F A257991(a(n)) = A350847(a(n)).

%e The terms together with their prime indices begin:

%e      1: ()

%e      6: (2,1)

%e     84: (4,2,1,1)

%e    126: (4,2,2,1)

%e    140: (4,3,1,1)

%e    210: (4,3,2,1)

%e    490: (4,4,3,1)

%e    525: (4,3,3,2)

%e    686: (4,4,4,1)

%e    875: (4,3,3,3)

%e   1404: (6,2,2,2,1,1)

%e   1456: (6,4,1,1,1,1)

%e   2106: (6,2,2,2,2,1)

%e   2184: (6,4,2,1,1,1)

%e   2288: (6,5,1,1,1,1)

%e   2340: (6,3,2,2,1,1)

%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];

%t Select[Range[1000],Count[primeMS[#],_?EvenQ]==Count[conj[primeMS[#]],_?OddQ]&&Count[primeMS[#],_?OddQ]==Count[conj[primeMS[#]],_?EvenQ]&]

%Y The first condition alone is A349157, counted by A277579.

%Y The second condition alone is A350943, counted by A277579.

%Y There are two other possible double-pairings of statistics:

%Y - A350946, counted by A351977.

%Y - A350949, counted by A351976.

%Y The case of all four statistics equal is A350947, counted by A351978.

%Y These partitions are counted by A351981.

%Y Partitions with as many even as odd parts:

%Y - counted by A045931

%Y - strict case counted by A239241

%Y - ranked by A325698

%Y - conjugate ranked by A350848

%Y - strict conjugate case counted by A352129

%Y A056239 adds up prime indices, counted by A001222, row sums of A112798.

%Y A122111 represents partition conjugation using Heinz numbers.

%Y A195017 = # of even parts - # of odd parts.

%Y A257991 counts odd parts, conjugate A344616.

%Y A257992 counts even parts, conjugate A350847.

%Y A316524 = alternating sum of prime indices.

%Y A350944: # of odd parts = # of odd conjugate parts, counted by A277103.

%Y A350945: # of even parts = # of even conjugate parts, counted by A350948.

%Y Cf. A026424, A028260, A098123, A130780, A171966, A241638, A325700, A350841, A350849, A350941, A350942, A350950, A350951.

%K nonn

%O 1,2

%A _Gus Wiseman_, Mar 14 2022